Documentation

Mathlib.Algebra.Order.Monoid.Unbundled.WithTop

Adjoining top/bottom elements to ordered monoids. #

instance WithTop.one {α : Type u} [One α] :
Equations
instance WithTop.zero {α : Type u} [Zero α] :
Equations
@[simp]
theorem WithTop.coe_one {α : Type u} [One α] :
1 = 1
@[simp]
theorem WithTop.coe_zero {α : Type u} [Zero α] :
0 = 0
@[simp]
theorem WithTop.coe_eq_one {α : Type u} [One α] {a : α} :
a = 1 a = 1
@[simp]
theorem WithTop.coe_eq_zero {α : Type u} [Zero α] {a : α} :
a = 0 a = 0
@[simp]
theorem WithTop.one_eq_coe {α : Type u} [One α] {a : α} :
1 = a a = 1
@[simp]
theorem WithTop.zero_eq_coe {α : Type u} [Zero α] {a : α} :
0 = a a = 0
@[simp]
theorem WithTop.top_ne_one {α : Type u} [One α] :
@[simp]
theorem WithTop.top_ne_zero {α : Type u} [Zero α] :
@[simp]
theorem WithTop.one_ne_top {α : Type u} [One α] :
@[simp]
theorem WithTop.zero_ne_top {α : Type u} [Zero α] :
@[simp]
theorem WithTop.untop_one {α : Type u} [One α] :
untop 1 = 1
@[simp]
theorem WithTop.untop_zero {α : Type u} [Zero α] :
untop 0 = 0
@[simp]
theorem WithTop.untopD_one {α : Type u} [One α] (d : α) :
untopD d 1 = 1
@[simp]
theorem WithTop.untopD_zero {α : Type u} [Zero α] (d : α) :
untopD d 0 = 0
@[deprecated WithTop.untopD_zero (since := "2025-02-06")]
theorem WithTop.untop_zero' {α : Type u} [Zero α] (d : α) :
untopD d 0 = 0

Alias of WithTop.untopD_zero.

@[deprecated WithTop.untopD_one (since := "2025-02-06")]
theorem WithTop.untop_one' {α : Type u} [One α] (d : α) :
untopD d 1 = 1

Alias of WithTop.untopD_one.

@[simp]
theorem WithTop.one_le_coe {α : Type u} [One α] [LE α] {a : α} :
1 a 1 a
@[simp]
theorem WithTop.coe_nonneg {α : Type u} [Zero α] [LE α] {a : α} :
0 a 0 a
@[simp]
theorem WithTop.coe_le_one {α : Type u} [One α] [LE α] {a : α} :
a 1 a 1
@[simp]
theorem WithTop.coe_le_zero {α : Type u} [Zero α] [LE α] {a : α} :
a 0 a 0
@[simp]
theorem WithTop.one_lt_coe {α : Type u} [One α] [LT α] {a : α} :
1 < a 1 < a
@[simp]
theorem WithTop.coe_pos {α : Type u} [Zero α] [LT α] {a : α} :
0 < a 0 < a
@[simp]
theorem WithTop.coe_lt_one {α : Type u} [One α] [LT α] {a : α} :
a < 1 a < 1
@[simp]
theorem WithTop.coe_lt_zero {α : Type u} [Zero α] [LT α] {a : α} :
a < 0 a < 0
@[simp]
theorem WithTop.map_one {α : Type u} [One α] {β : Type u_1} (f : αβ) :
map f 1 = (f 1)
@[simp]
theorem WithTop.map_zero {α : Type u} [Zero α] {β : Type u_1} (f : αβ) :
map f 0 = (f 0)
theorem WithTop.map_eq_one_iff {β : Type v} {α : Type u_1} {f : αβ} {v : WithTop α} [One β] :
map f v = 1 (x : α), v = x f x = 1
theorem WithTop.map_eq_zero_iff {β : Type v} {α : Type u_1} {f : αβ} {v : WithTop α} [Zero β] :
map f v = 0 (x : α), v = x f x = 0
theorem WithTop.one_eq_map_iff {β : Type v} {α : Type u_1} {f : αβ} {v : WithTop α} [One β] :
1 = map f v (x : α), v = x f x = 1
theorem WithTop.zero_eq_map_iff {β : Type v} {α : Type u_1} {f : αβ} {v : WithTop α} [Zero β] :
0 = map f v (x : α), v = x f x = 0
instance WithTop.zeroLEOneClass {α : Type u} [One α] [Zero α] [LE α] [ZeroLEOneClass α] :
instance WithTop.add {α : Type u} [Add α] :
Equations
@[simp]
theorem WithTop.coe_add {α : Type u} [Add α] (a b : α) :
↑(a + b) = a + b
@[simp]
theorem WithTop.top_add {α : Type u} [Add α] (x : WithTop α) :
@[simp]
theorem WithTop.add_top {α : Type u} [Add α] (x : WithTop α) :
@[simp]
theorem WithTop.add_eq_top {α : Type u} [Add α] {x y : WithTop α} :
x + y = x = y =
theorem WithTop.add_ne_top {α : Type u} [Add α] {x y : WithTop α} :
@[simp]
theorem WithTop.add_lt_top {α : Type u} [Add α] {x y : WithTop α} [LT α] :
x + y < x < y <
theorem WithTop.add_eq_coe {α : Type u} [Add α] {a b : WithTop α} {c : α} :
a + b = c (a' : α), (b' : α), a' = a b' = b a' + b' = c
theorem WithTop.add_coe_eq_top_iff {α : Type u} [Add α] {x : WithTop α} {b : α} :
x + b = x =
theorem WithTop.coe_add_eq_top_iff {α : Type u} [Add α] {y : WithTop α} {a : α} :
a + y = y =
theorem WithTop.add_right_inj {α : Type u} [Add α] {x y z : WithTop α} [IsRightCancelAdd α] (hz : z ) :
x + z = y + z x = y
theorem WithTop.add_right_cancel {α : Type u} [Add α] {x y z : WithTop α} [IsRightCancelAdd α] (hz : z ) (h : x + z = y + z) :
x = y
theorem WithTop.add_left_inj {α : Type u} [Add α] {x y z : WithTop α} [IsLeftCancelAdd α] (hx : x ) :
x + y = x + z y = z
theorem WithTop.add_left_cancel {α : Type u} [Add α] {x y z : WithTop α} [IsLeftCancelAdd α] (hx : x ) (h : x + y = x + z) :
y = z
@[deprecated WithTop.add_left_inj (since := "2025-02-19")]
theorem WithTop.add_left_cancel_iff {α : Type u} [Add α] {x y z : WithTop α} [IsLeftCancelAdd α] (hx : x ) :
x + y = x + z y = z

Alias of WithTop.add_left_inj.

@[deprecated WithTop.add_right_inj (since := "2025-02-19")]
theorem WithTop.add_right_cancel_iff {α : Type u} [Add α] {x y z : WithTop α} [IsRightCancelAdd α] (hz : z ) :
x + z = y + z x = y

Alias of WithTop.add_right_inj.

instance WithTop.addLeftMono {α : Type u} [Add α] [LE α] [AddLeftMono α] :
instance WithTop.addRightMono {α : Type u} [Add α] [LE α] [AddRightMono α] :
theorem WithTop.le_of_add_le_add_left {α : Type u} [Add α] {x y z : WithTop α} [LE α] [AddLeftReflectLE α] (hx : x ) :
x + y x + zy z
theorem WithTop.le_of_add_le_add_right {α : Type u} [Add α] {x y z : WithTop α} [LE α] [AddRightReflectLE α] (hz : z ) :
x + z y + zx y
theorem WithTop.add_lt_add_left {α : Type u} [Add α] {x y z : WithTop α} [LT α] [AddLeftStrictMono α] (hx : x ) :
y < zx + y < x + z
theorem WithTop.add_lt_add_right {α : Type u} [Add α] {x y z : WithTop α} [LT α] [AddRightStrictMono α] (hz : z ) :
x < yx + z < y + z
theorem WithTop.add_le_add_iff_left {α : Type u} [Add α] {x y z : WithTop α} [LE α] [AddLeftMono α] [AddLeftReflectLE α] (hx : x ) :
x + y x + z y z
theorem WithTop.add_le_add_iff_right {α : Type u} [Add α] {x y z : WithTop α} [LE α] [AddRightMono α] [AddRightReflectLE α] (hz : z ) :
x + z y + z x y
theorem WithTop.add_lt_add_iff_left {α : Type u} [Add α] {x y z : WithTop α} [LT α] [AddLeftStrictMono α] [AddLeftReflectLT α] (hx : x ) :
x + y < x + z y < z
theorem WithTop.add_lt_add_iff_right {α : Type u} [Add α] {x y z : WithTop α} [LT α] [AddRightStrictMono α] [AddRightReflectLT α] (hz : z ) :
x + z < y + z x < y
theorem WithTop.add_lt_add_of_le_of_lt {α : Type u} [Add α] {w x y z : WithTop α} [Preorder α] [AddLeftStrictMono α] [AddRightMono α] (hw : w ) (hwy : w y) (hxz : x < z) :
w + x < y + z
theorem WithTop.add_lt_add_of_lt_of_le {α : Type u} [Add α] {w x y z : WithTop α} [Preorder α] [AddLeftMono α] [AddRightStrictMono α] (hx : x ) (hwy : w < y) (hxz : x z) :
w + x < y + z
theorem WithTop.addLECancellable_of_ne_top {α : Type u} [Add α] {x : WithTop α} [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] (hx : x ) :
theorem WithTop.addLECancellable_of_lt_top {α : Type u} [Add α] {x : WithTop α} [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] (hx : x < ) :
theorem WithTop.addLECancellable_coe {α : Type u} [Add α] [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] (a : α) :
theorem WithTop.addLECancellable_iff_ne_top {α : Type u} [Add α] {x : WithTop α} [Nonempty α] [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] :
@[simp]
theorem WithTop.map_add {α : Type u} {β : Type v} [Add α] {F : Type u_1} [Add β] [FunLike F α β] [AddHomClass F α β] (f : F) (a b : WithTop α) :
map (⇑f) (a + b) = map (⇑f) a + map (⇑f) b
instance WithTop.addMonoid {α : Type u} [AddMonoid α] :
Equations
@[simp]
theorem WithTop.coe_nsmul {α : Type u} [AddMonoid α] (a : α) (n : ) :
↑(n a) = n a
def WithTop.addHom {α : Type u} [AddMonoid α] :

Coercion from α to WithTop α as an AddMonoidHom.

Equations
@[simp]
theorem WithTop.coe_addHom {α : Type u} [AddMonoid α] :
@[simp]
theorem WithTop.coe_natCast {α : Type u} [AddMonoidWithOne α] (n : ) :
n = n
@[simp]
theorem WithTop.top_ne_natCast {α : Type u} [AddMonoidWithOne α] (n : ) :
n
@[simp]
theorem WithTop.natCast_ne_top {α : Type u} [AddMonoidWithOne α] (n : ) :
n
@[simp]
theorem WithTop.natCast_lt_top {α : Type u} [AddMonoidWithOne α] [LT α] (n : ) :
n <
@[simp]
theorem WithTop.coe_ofNat {α : Type u} [AddMonoidWithOne α] (n : ) [n.AtLeastTwo] :
@[simp]
theorem WithTop.coe_eq_ofNat {α : Type u} [AddMonoidWithOne α] (n : ) [n.AtLeastTwo] (m : α) :
@[simp]
theorem WithTop.ofNat_eq_coe {α : Type u} [AddMonoidWithOne α] (n : ) [n.AtLeastTwo] (m : α) :
@[simp]
theorem WithTop.map_ofNat {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : αβ} (n : ) [n.AtLeastTwo] :
map f (OfNat.ofNat n) = (f (OfNat.ofNat n))
@[simp]
theorem WithTop.map_natCast {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : αβ} (n : ) :
map f n = (f n)
theorem WithTop.map_eq_ofNat_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : βα} {n : } [n.AtLeastTwo] {a : WithTop β} :
map f a = OfNat.ofNat n (x : β), a = x f x = n
theorem WithTop.ofNat_eq_map_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : βα} {n : } [n.AtLeastTwo] {a : WithTop β} :
OfNat.ofNat n = map f a (x : β), a = x f x = n
theorem WithTop.map_eq_natCast_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : βα} {n : } {a : WithTop β} :
map f a = n (x : β), a = x f x = n
theorem WithTop.natCast_eq_map_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : βα} {n : } {a : WithTop β} :
n = map f a (x : β), a = x f x = n
@[simp]
theorem WithTop.one_lt_top {α : Type u} [One α] [LT α] :
1 <
@[simp]
theorem WithTop.top_pos {α : Type u} [Zero α] [LT α] :
0 <
@[deprecated WithTop.top_pos (since := "2024-10-22")]
theorem WithTop.zero_lt_top {α : Type u} [Zero α] [LT α] :
0 <

Alias of WithTop.top_pos.

@[deprecated WithTop.coe_pos (since := "2024-10-22")]
theorem WithTop.zero_lt_coe {α : Type u} [Zero α] [LT α] (a : α) :
0 < a 0 < a
def OneHom.withTopMap {M : Type u_1} {N : Type u_2} [One M] [One N] (f : OneHom M N) :

A version of WithTop.map for OneHoms.

Equations
def ZeroHom.withTopMap {M : Type u_1} {N : Type u_2} [Zero M] [Zero N] (f : ZeroHom M N) :

A version of WithTop.map for ZeroHoms

Equations
@[simp]
theorem OneHom.withTopMap_apply {M : Type u_1} {N : Type u_2} [One M] [One N] (f : OneHom M N) :
@[simp]
theorem ZeroHom.withTopMap_apply {M : Type u_1} {N : Type u_2} [Zero M] [Zero N] (f : ZeroHom M N) :
def AddHom.withTopMap {M : Type u_1} {N : Type u_2} [Add M] [Add N] (f : M →ₙ+ N) :

A version of WithTop.map for AddHoms.

Equations
@[simp]
theorem AddHom.withTopMap_apply {M : Type u_1} {N : Type u_2} [Add M] [Add N] (f : M →ₙ+ N) :
def AddMonoidHom.withTopMap {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) :

A version of WithTop.map for AddMonoidHoms.

Equations
@[simp]
theorem AddMonoidHom.withTopMap_apply {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) :
instance WithBot.one {α : Type u} [One α] :
Equations
instance WithBot.zero {α : Type u} [Zero α] :
Equations
@[simp]
theorem WithBot.coe_one {α : Type u} [One α] :
1 = 1
@[simp]
theorem WithBot.coe_zero {α : Type u} [Zero α] :
0 = 0
@[simp]
theorem WithBot.coe_eq_one {α : Type u} [One α] {a : α} :
a = 1 a = 1
@[simp]
theorem WithBot.coe_eq_zero {α : Type u} [Zero α] {a : α} :
a = 0 a = 0
@[simp]
theorem WithBot.one_eq_coe {α : Type u} [One α] {a : α} :
1 = a a = 1
@[simp]
theorem WithBot.zero_eq_coe {α : Type u} [Zero α] {a : α} :
0 = a a = 0
@[simp]
theorem WithBot.bot_ne_one {α : Type u} [One α] :
@[simp]
theorem WithBot.bot_ne_zero {α : Type u} [Zero α] :
@[simp]
theorem WithBot.one_ne_bot {α : Type u} [One α] :
@[simp]
theorem WithBot.zero_ne_bot {α : Type u} [Zero α] :
@[simp]
theorem WithBot.unbot_one {α : Type u} [One α] :
unbot 1 = 1
@[simp]
theorem WithBot.unbot_zero {α : Type u} [Zero α] :
unbot 0 = 0
@[simp]
theorem WithBot.unbotD_one {α : Type u} [One α] (d : α) :
unbotD d 1 = 1
@[simp]
theorem WithBot.unbotD_zero {α : Type u} [Zero α] (d : α) :
unbotD d 0 = 0
@[deprecated WithBot.unbotD_zero (since := "2025-02-06")]
theorem WithBot.unbot_zero' {α : Type u} [Zero α] (d : α) :
unbotD d 0 = 0

Alias of WithBot.unbotD_zero.

@[deprecated WithBot.unbotD_one (since := "2025-02-06")]
theorem WithBot.unbot_one' {α : Type u} [One α] (d : α) :
unbotD d 1 = 1

Alias of WithBot.unbotD_one.

@[simp]
theorem WithBot.one_le_coe {α : Type u} [One α] {a : α} [LE α] :
1 a 1 a
@[simp]
theorem WithBot.coe_nonneg {α : Type u} [Zero α] {a : α} [LE α] :
0 a 0 a
@[simp]
theorem WithBot.coe_le_one {α : Type u} [One α] {a : α} [LE α] :
a 1 a 1
@[simp]
theorem WithBot.coe_le_zero {α : Type u} [Zero α] {a : α} [LE α] :
a 0 a 0
@[simp]
theorem WithBot.one_lt_coe {α : Type u} [One α] {a : α} [LT α] :
1 < a 1 < a
@[simp]
theorem WithBot.coe_pos {α : Type u} [Zero α] {a : α} [LT α] :
0 < a 0 < a
@[simp]
theorem WithBot.coe_lt_one {α : Type u} [One α] {a : α} [LT α] :
a < 1 a < 1
@[simp]
theorem WithBot.coe_lt_zero {α : Type u} [Zero α] {a : α} [LT α] :
a < 0 a < 0
@[simp]
theorem WithBot.map_one {α : Type u} [One α] {β : Type u_1} (f : αβ) :
map f 1 = (f 1)
@[simp]
theorem WithBot.map_zero {α : Type u} [Zero α] {β : Type u_1} (f : αβ) :
map f 0 = (f 0)
theorem WithBot.map_eq_one_iff {β : Type v} {α : Type u_1} {f : αβ} {v : WithBot α} [One β] :
map f v = 1 (x : α), v = x f x = 1
theorem WithBot.map_eq_zero_iff {β : Type v} {α : Type u_1} {f : αβ} {v : WithBot α} [Zero β] :
map f v = 0 (x : α), v = x f x = 0
theorem WithBot.one_eq_map_iff {β : Type v} {α : Type u_1} {f : αβ} {v : WithBot α} [One β] :
1 = map f v (x : α), v = x f x = 1
theorem WithBot.zero_eq_map_iff {β : Type v} {α : Type u_1} {f : αβ} {v : WithBot α} [Zero β] :
0 = map f v (x : α), v = x f x = 0
instance WithBot.zeroLEOneClass {α : Type u} [One α] [Zero α] [LE α] [ZeroLEOneClass α] :
instance WithBot.add {α : Type u} [Add α] :
Equations
@[simp]
theorem WithBot.coe_add {α : Type u} [Add α] (a b : α) :
↑(a + b) = a + b
@[simp]
theorem WithBot.bot_add {α : Type u} [Add α] (x : WithBot α) :
@[simp]
theorem WithBot.add_bot {α : Type u} [Add α] (x : WithBot α) :
@[simp]
theorem WithBot.add_eq_bot {α : Type u} [Add α] {x y : WithBot α} :
x + y = x = y =
theorem WithBot.add_ne_bot {α : Type u} [Add α] {x y : WithBot α} :
@[simp]
theorem WithBot.bot_lt_add {α : Type u} [Add α] {x y : WithBot α} [LT α] :
< x + y < x < y
theorem WithBot.add_eq_coe {α : Type u} [Add α] {a b : WithBot α} {c : α} :
a + b = c (a' : α), (b' : α), a' = a b' = b a' + b' = c
theorem WithBot.add_coe_eq_bot_iff {α : Type u} [Add α] {x : WithBot α} {b : α} :
x + b = x =
theorem WithBot.coe_add_eq_bot_iff {α : Type u} [Add α] {y : WithBot α} {a : α} :
a + y = y =
theorem WithBot.add_right_inj {α : Type u} [Add α] {x y z : WithBot α} [IsRightCancelAdd α] (hz : z ) :
x + z = y + z x = y
theorem WithBot.add_right_cancel {α : Type u} [Add α] {x y z : WithBot α} [IsRightCancelAdd α] (hz : z ) (h : x + z = y + z) :
x = y
theorem WithBot.add_left_inj {α : Type u} [Add α] {x y z : WithBot α} [IsLeftCancelAdd α] (hx : x ) :
x + y = x + z y = z
theorem WithBot.add_left_cancel {α : Type u} [Add α] {x y z : WithBot α} [IsLeftCancelAdd α] (hx : x ) (h : x + y = x + z) :
y = z
@[deprecated WithBot.add_left_inj (since := "2025-02-19")]
theorem WithBot.add_left_cancel_iff {α : Type u} [Add α] {x y z : WithBot α} [IsLeftCancelAdd α] (hx : x ) :
x + y = x + z y = z

Alias of WithBot.add_left_inj.

@[deprecated WithBot.add_right_inj (since := "2025-02-19")]
theorem WithBot.add_right_cancel_iff {α : Type u} [Add α] {x y z : WithBot α} [IsRightCancelAdd α] (hz : z ) :
x + z = y + z x = y

Alias of WithBot.add_right_inj.

instance WithBot.addLeftMono {α : Type u} [Add α] [LE α] [AddLeftMono α] :
instance WithBot.addRightMono {α : Type u} [Add α] [LE α] [AddRightMono α] :
theorem WithBot.le_of_add_le_add_left {α : Type u} [Add α] {x y z : WithBot α} [LE α] [AddLeftReflectLE α] (hx : x ) :
x + y x + zy z
theorem WithBot.le_of_add_le_add_right {α : Type u} [Add α] {x y z : WithBot α} [LE α] [AddRightReflectLE α] (hz : z ) :
x + z y + zx y
theorem WithBot.add_lt_add_left {α : Type u} [Add α] {x y z : WithBot α} [LT α] [AddLeftStrictMono α] (hx : x ) :
y < zx + y < x + z
theorem WithBot.add_lt_add_right {α : Type u} [Add α] {x y z : WithBot α} [LT α] [AddRightStrictMono α] (hz : z ) :
x < yx + z < y + z
theorem WithBot.add_le_add_iff_left {α : Type u} [Add α] {x y z : WithBot α} [LE α] [AddLeftMono α] [AddLeftReflectLE α] (hx : x ) :
x + y x + z y z
theorem WithBot.add_le_add_iff_right {α : Type u} [Add α] {x y z : WithBot α} [LE α] [AddRightMono α] [AddRightReflectLE α] (hz : z ) :
x + z y + z x y
theorem WithBot.add_lt_add_iff_left {α : Type u} [Add α] {x y z : WithBot α} [LT α] [AddLeftStrictMono α] [AddLeftReflectLT α] (hx : x ) :
x + y < x + z y < z
theorem WithBot.add_lt_add_iff_right {α : Type u} [Add α] {x y z : WithBot α} [LT α] [AddRightStrictMono α] [AddRightReflectLT α] (hz : z ) :
x + z < y + z x < y
theorem WithBot.add_lt_add_of_le_of_lt {α : Type u} [Add α] {w x y z : WithBot α} [Preorder α] [AddLeftStrictMono α] [AddRightMono α] (hw : w ) (hwy : w y) (hxz : x < z) :
w + x < y + z
theorem WithBot.add_lt_add_of_lt_of_le {α : Type u} [Add α] {w x y z : WithBot α} [Preorder α] [AddLeftMono α] [AddRightStrictMono α] (hx : x ) (hwy : w < y) (hxz : x z) :
w + x < y + z
theorem WithBot.addLECancellable_of_ne_bot {α : Type u} [Add α] {x : WithBot α} [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] (hx : x ) :
theorem WithBot.addLECancellable_of_lt_bot {α : Type u} [Add α] {x : WithBot α} [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] (hx : x < ) :
theorem WithBot.addLECancellable_coe {α : Type u} [Add α] [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] (a : α) :
theorem WithBot.addLECancellable_iff_ne_bot {α : Type u} [Add α] {x : WithBot α} [Nonempty α] [Preorder α] [ContravariantClass α α (fun (x1 x2 : α) => x1 + x2) fun (x1 x2 : α) => x1 x2] :
@[simp]
theorem WithBot.map_add {α : Type u} {β : Type v} [Add α] {F : Type u_1} [Add β] [FunLike F α β] [AddHomClass F α β] (f : F) (a b : WithBot α) :
map (⇑f) (a + b) = map (⇑f) a + map (⇑f) b
def WithBot.addHom {α : Type u} [AddMonoid α] :

Coercion from α to WithBot α as an AddMonoidHom.

Equations
@[simp]
theorem WithBot.coe_addHom {α : Type u} [AddMonoid α] :
@[simp]
theorem WithBot.coe_nsmul {α : Type u} [AddMonoid α] (a : α) (n : ) :
↑(n a) = n a
theorem WithBot.coe_natCast {α : Type u} [AddMonoidWithOne α] (n : ) :
n = n
@[simp]
theorem WithBot.natCast_ne_bot {α : Type u} [AddMonoidWithOne α] (n : ) :
n
@[simp]
theorem WithBot.bot_ne_natCast {α : Type u} [AddMonoidWithOne α] (n : ) :
n
@[simp]
theorem WithBot.coe_ofNat {α : Type u} [AddMonoidWithOne α] (n : ) [n.AtLeastTwo] :
@[simp]
theorem WithBot.coe_eq_ofNat {α : Type u} [AddMonoidWithOne α] (n : ) [n.AtLeastTwo] (m : α) :
@[simp]
theorem WithBot.ofNat_eq_coe {α : Type u} [AddMonoidWithOne α] (n : ) [n.AtLeastTwo] (m : α) :
@[simp]
theorem WithBot.map_ofNat {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : αβ} (n : ) [n.AtLeastTwo] :
map f (OfNat.ofNat n) = (f (OfNat.ofNat n))
@[simp]
theorem WithBot.map_natCast {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : αβ} (n : ) :
map f n = (f n)
theorem WithBot.map_eq_ofNat_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : βα} {n : } [n.AtLeastTwo] {a : WithBot β} :
map f a = OfNat.ofNat n (x : β), a = x f x = n
theorem WithBot.ofNat_eq_map_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : βα} {n : } [n.AtLeastTwo] {a : WithBot β} :
OfNat.ofNat n = map f a (x : β), a = x f x = n
theorem WithBot.map_eq_natCast_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : βα} {n : } {a : WithBot β} :
map f a = n (x : β), a = x f x = n
theorem WithBot.natCast_eq_map_iff {α : Type u} {β : Type v} [AddMonoidWithOne α] {f : βα} {n : } {a : WithBot β} :
n = map f a (x : β), a = x f x = n
def OneHom.withBotMap {M : Type u_1} {N : Type u_2} [One M] [One N] (f : OneHom M N) :

A version of WithBot.map for OneHoms.

Equations
def ZeroHom.withBotMap {M : Type u_1} {N : Type u_2} [Zero M] [Zero N] (f : ZeroHom M N) :

A version of WithBot.map for ZeroHoms

Equations
@[simp]
theorem ZeroHom.withBotMap_apply {M : Type u_1} {N : Type u_2} [Zero M] [Zero N] (f : ZeroHom M N) :
@[simp]
theorem OneHom.withBotMap_apply {M : Type u_1} {N : Type u_2} [One M] [One N] (f : OneHom M N) :
def AddHom.withBotMap {M : Type u_1} {N : Type u_2} [Add M] [Add N] (f : M →ₙ+ N) :

A version of WithBot.map for AddHoms.

Equations
@[simp]
theorem AddHom.withBotMap_apply {M : Type u_1} {N : Type u_2} [Add M] [Add N] (f : M →ₙ+ N) :
def AddMonoidHom.withBotMap {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) :

A version of WithBot.map for AddMonoidHoms.

Equations
@[simp]
theorem AddMonoidHom.withBotMap_apply {M : Type u_1} {N : Type u_2} [AddZeroClass M] [AddZeroClass N] (f : M →+ N) :